Problem 19 - Fundamentals of Physics

Question

What is the resistivity of a wire of \(1.0 \mathrm{~mm}\) diameter, \(2.0 \mathrm{~m}\) length, and \(50 \mathrm{~m} \Omega\) resistance?

Step-by-Step Solution

Answer

The resistivity is \(1.9625 \times 10^{-8} \Omega \cdot \text{m}\).

Step 1: Understand the Given Values

First, let's understand the values given in the problem. We have the diameter of the wire as \(1.0 \text{ mm}\), which can be converted to meters by dividing by 1000, giving us \(0.001 \text{ m}\). The length of the wire is given as \(2.0 \text{ m}\). The resistance of the wire is \(50 \text{ m}\Omega\), which is equivalent to \(0.050 \Omega\).

Step 2: Relate Resistivity Formula

The formula to calculate resistivity \(\rho\) is given by \(\rho = R \frac{A}{L}\), where \(R\) is the resistance, \(A\) is the cross-sectional area, and \(L\) is the length of the wire.

Step 3: Calculate Cross-Sectional Area

The cross-sectional area \(A\) of the wire can be calculated using the formula for the area of a circle, \(A = \pi r^2\), where \(r\) is the radius. The radius \(r\) is half of the diameter: \[ r = \frac{1.0 \text{ mm}}{2} = \frac{0.001 \text{ m}}{2} = 0.0005 \text{ m} \]. \[ A = \pi (0.0005)^2 = \pi \times 0.00000025 = 7.85 \times 10^{-7} \text{ m}^2. \]

Step 4: Substitute Values into Formula

Now, substitute the values into the resistivity formula: \[ \rho = (0.050) \frac{7.85 \times 10^{-7}}{2.0} \].

Step 5: Calculate and Interpret

Complete the calculation: \[ \rho = (0.050) \times 3.925 \times 10^{-7} = 1.9625 \times 10^{-8} \Omega \cdot \text{m} \]. Thus, the resistivity of the wire is \(1.9625 \times 10^{-8} \Omega \cdot \text{m}\).

Key Concepts

Resistance

Resistance is a fundamental concept in understanding how electrical current flows through a material. Resistance, denoted by the symbol \( R \), determines how much a material opposes the flow of electric current. It is measured in ohms (\( \Omega \)). The resistance of a wire depends on several factors:

• The resistivity of the material, which is an intrinsic property.
• The length of the wire: longer wires have greater resistance.
• The cross-sectional area: thinner wires have more resistance.

In the given problem, the resistance is provided as \(50 \text{ m} \Omega\), which is equivalent to \(0.050 \Omega\). This acts as a barrier to the free flow of electrons, affecting how effectively electricity is conducted.

Cross-Sectional Area

The cross-sectional area of a wire is crucial as it impacts the resistance. A larger cross-sectional area allows more electrons to flow, thereby reducing resistance.
Consider the cross-section of a wire to be circular. Thus, we can use the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius.
In the exercise, the diameter is given as \(1.0 \text{ mm}\), which first needs to be converted to meters: \(0.001 \text{ m}\). The radius is half the diameter, hence \(0.0005 \text{ m}\). Plugging this value into the formula, we calculated:

• \( A = \pi (0.0005)^2 = 7.85 \times 10^{-7} \text{ m}^2 \).

This shows a small cross-sectional area, which will have a higher resistance if the material were to remain the same size.

Resistivity Formula

The resistivity formula is pivotal in solving problems related to wire resistance. The formula is given by:\[\rho = R \frac{A}{L}\]where:

• \( \rho \) is the resistivity, measured in ohm-meters (\( \Omega \cdot \text{m} \)).
• \( R \) is the resistance.
• \( A \) is the cross-sectional area.
• \( L \) is the length of the wire.

This formula connects how intrinsic the property of the wire, resistivity, relates with tangible measurements like resistance, area, and length. In practice, knowing any of these three values can help compute the resistivity, which remains constant for a given temperature and material. It indicates how resistive a material inherently is to the current flow.

Length of Wire

The length of a wire is another critical factor that affects its resistance. A longer wire means electrons have to travel a greater distance and are more likely to encounter obstacles that impede movement. This leads to higher resistance.
In the exercise, the wire's length is given as \(2.0 \text{ m}\). Doubling the length, for instance, would roughly double the resistance, assuming all other factors remained constant. Hence, the relationship between resistivity and length is direct: the length increases, and so does the resistance.
This understanding helps engineers and physicists design circuits and cables appropriately for the necessary electrical requirements.